Spiric section

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r=2, a=1, c=0, 0.8, 1

A spiric section is a special case of a toric section, which is the intersection of a plane with a torus (σπειρα in ancient Greek). Spiric sections are toric sections in which the intersecting plane is parallel to the rotational symmetry axis of the torus. Spiric sections were discovered by the ancient Greek geometer Perseus in roughly 150 BC, and are assumed to be the first toric sections to be described.

[edit] Mathematical description

In general, spiric sections are fourth-order (quartic) plane curves with three parameters

$\left( r^{2} - a^{2} + c^{2} + x^{2} + y^{2} \right)^{2} = 4r^{2} \left( x^{2} + c^{2} \right)$

In this formula, the torus is formed by rotating a circle of radius $a$ with its center following another circle of radius $r$ (not necessarily larger than $a$ , self-intersection is OK). The parameter $c$ is the shortest distance from the intersecting plane to the (parallel) rotational symmetry axis. There are no spiric sections with $c > r + a$ , since there is no intersection; the plane is too far away from the torus to intersect it.

The overall scale dependence can be eliminated by setting $r = 1$ . Such normal-form spiric sections are a 2-parameter family of quartic plane curves.

[edit] Examples of spiric sections

Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli. The Cassini oval has the remarkable property that the product of distances to two foci are constant. For comparison, the sum is constant in ellipses, the difference is constant in hyperbolae and the ratio is constant in circles.